General relativity from string theory

[Haelfix]

"The perturbations are imagined to change the geometry but they continue to run on a pre-established web of causality."

No they do not! Consider the case classically.

For a weak field expansion G --> sum (1..n) G0 + H1 + H2 _ ... where each piece of H i, is required to couple to its own stress energy tensor, the lines of causality seem to follow that which is set down by G0.

But this is a mirage! Consider a change in the perturbation H1+ e J1 + ... (where e is a small epsilon) Linearing the action around G0 + H1 + e Ji + ... and throwing out terms with more than 1 powers of H, yields the result that j propagates not on the light cone of G0, but rather on the new light cone G0 + H1.

Every successive order, adds an extra term. In other words the lightcone converges not to the background G0, but to the lightcone fixed by the 'order by order improved action'. Summing up infinitely many of these terms yields the full nonlinear field equations, just like Deser/Weinberg/Feynman et al showed in their classic papers. In fact, I think there is a clever way to terminate the series after only a few terms.

No the more problematic issue is not causality, its that you might have global topological ambiguities with the weak field expansion, and its not at all obvious how you go about deciding which topology you are in when you are working in that formalism.

 

[Sardano]

And you should check Gravity and Strings by Tomas Ortin, chapter 3, where he recovers GR as SRFT.

 

[petergreat]

String theory predicts lots of modifications to the Einstein-Hilbert action, including the presence of the dilaton, the antisymmetric tensor, and higher derivative terms. Why is it the case that gravity in the real world is almost perfectly described by the Einstein-Hilbert action?

 

[Sardano]

In the low energy approximation you get GR. But of course ST predicts corrections away from this limit. The usual GR is not such a "perfect" description of the universe: it has been only through some local tests in the solar system and at cosmological scales is expected to be corrected: in fact, there are plenty of people working on modifications of EH action who are not ST physicist. I am talking about f(R) models, vector -tensor theories of gravity...etc etc some of them with great experimental results:

http://arxiv.org/abs/0904.0433

http://arxiv.org/abs/0905.1245

Besides, there are several "metric theories of gravity" which gives the same results as the GR.
It is clear that the EH term should be corrected. The LQG people are the only ones that do not understand this fact.

 

[atyy]

http://relativity.livingreviews.org/Articles/lrr-2000-5/ , section 3.3.2 and ref. 42

It seems that harmonic coordinates can in fact penetrate the event horizon, contrary to my expectations in post #33, and consistent with finbar's statement in post #32 that the Schwarzschild solution is well approximated as a coherent state of gravitons.

 

[Finbar]

I think the same goes for Schwarzschild dS/AdS(or more generally any Einstein space). Where you can view the gravitons as perturbations around dS/AdS "vacuum". The reason for this is that the Weyl curvature vanishes in dS/AdS and Weyl curvature is the propagating force which gravitons carry. Of coarse in string theory one needs to find solutions that give AdS/dS first. For AdS these seem to be provided by AdS/CFT at least.

 

[tom.stoer]

It is clear that the EH term should be corrected. The LQG people are the only ones that do not understand this fact.

They are perfectly aware of it.

LQG is not simply about quantizing one specific action, it is about a new method of quantizing gravity and there is no reason why this method should be restricted to just the EH action. Of course there are quantum corrections to the EH action which is no longer the central object in the theory but which is replaced by spin networks and which should arise only as a certain semiclassical approximation.

 

[Finbar]

Is there something like an effective action in spin foams/LQG? A generating functional that contains all quantum corrections??

 

[qsa]

GR does not predict the expansion of the universe without the clumsy CC which is an after thought. the same problem with all QG theories which tries to justify CC with all kinds of tricks. That shows that string mostly is not a fundamental theory.

 

[atyy]

Is there something like an effective action in spin foams/LQG? A generating functional that contains all quantum corrections??

I don't think there is an effective action yet. However, there is a full theory which in a suitable limit/corase graining should gve you something like an effective action. The state of the art is http://arxiv.org/abs/0905.4082 , http://arxiv.org/abs/1103.4602 , http://arxiv.org/abs/1105.0216 , and http://arxiv.org/abs/1105.0566 . So far so good, but it is unclear if the full theory is convergent, whether the right limits have been taken, and even if it is the right limit, whether so many terms have been omitted that the divergence is not seen. Also, they don't know whether the correct limit is cos(iSR) or exp(iSR). All the above papers argue for the former, but this one seems to argue for the latter http://arxiv.org/abs/1004.4550

 

[tom.stoer]

Is there something like an effective action in spin foams/LQG? A generating functional that contains all quantum corrections??

The issue when comparing LQG and strings is not the effective action; LQG in spinfoam representation is closed to Kadanoffs blockspin picture (refer to ordinary QM & condensed matter physics), therefore arbitrary f(R) terms originating from the classical action and new couplings at vertices could be taken into account; the theory will be investigated w.r.t. its renormalization properties etc.

The main issue is how to take matter d.o.f. into account and how they will change the quantum properties of spacetime. In string theory geometry and matter are two properties of one single fundamental object whereas in LQG matter does not (yet) emerge from spacetime but is put in on top of it. This is a fundamental different picture and is therefore to a large degree arbitrary.

My expectation is that on long run LQG will only provide a reasonable theory of nature if it allows us to understand matter as an aspect of spacetime, not as something put in on top. This could e.g. be achieved via algebraic extensions (colorings) or via topological properties (braiding, knotting of graphs). Nevertheless LQG has much value as it provides a new way of quantization which is fundamentally non-perturbative, fully dynamical i.e. background independent from the very beginning.

 

[qsa]

The issue when comparing LQG and strings ...QUOTE]


tom, do you disagree with my last post, and if yes, why.

 

[tom.stoer]

GR does predict expansion; it simply does not predict accelerated expansion w/o the cc.

In a general picture like the asymptotic safety approach the cc is nothing else but one (afaik not very special) coupling constant. Therefore I expect that string theory should be able to deal with the cc as well, i.e. that besides AdS dS is required, too. In addition I think that any viable approach to QG should not use tricks for the cc and treat it is something special.

But I can't see how you can conclude that strings are not fundamental b/c of the cc. The fact that it's not perfectly understood in string theory does not spoil the whole concept.

 

[qsa]

GR does predict expansion; it simply does not predict accelerated expansion w/o the cc.

In a general picture like the asymptotic safety approach the cc is nothing else but one (afaik not very special) coupling constant. Therefore I expect that string theory should be able to deal with the cc as well, i.e. that besides AdS dS is required, too. In addition I think that any viable approach to QG should not use tricks for the cc and treat it is something special.

But I can't see how you can conclude that strings are not fundamental b/c of the cc. The fact that it's not perfectly understood in string theory does not spoil the whole concept.

Thanks. Of course, I did mean accelerated expansion. It is one thing for a general fundamental theory not to predict mass but to miss the behavior sound very disappointing. Even AS uses E-H action with modification, don't they all, more or less.

http://arxiv.org/pdf/1012.2680v1

I did not mean to say that string is not usefull, but just treating it as a theory that ends all was unjustified giving these relatively simple but fundamental issues. But your answer was good as always.

 

[Finbar]

qsa makes a good point. If we go by Tong's calculation string theory is only consistent in strictly Ricci flat space-times. Since evidence(accelerated expansion) points to us living in de-sitter space we must need to find some degrees of freedom(modes of the string), other than gravitons, which form a coherent state corresponding to de-sitter space.

Anyone know if this has been achieved??

 

[Finbar]

. Even AS uses E-H action with modification, don't they all, more or less.

To prove the AS conjecture every possible term must be included in the effective action.

So far evidence points to certain RG trajectories which do in fact resolve the cosmological constant problem. See http://arxiv.org/pdf/hep-th/0702051 page 10 for example.

 

[tom.stoer]

If we go by Tong's calculation string theory is only consistent in strictly Ricci flat space-times. Since evidence ... points to us living in de-sitter space we must need to find some degrees of freedom ... which form a coherent state corresponding to de-sitter space.

I guess this is a fundamental problem, namely that in a certain sense background independenca means something different in string theory. One has to prove for a certain background that a consistent quantization can be achieved. And this has to be done for each background seperately. Therefore the background (or let's say the class of backgrounds) changes the d.o.f.

 

[Finbar]

I don't know if i agree. It seems very plausible that one can simply think of minkowski space as the ground state of the theory. Remember that there is still a lot of structure in flat space-time(Unrhuh temperature etc.) and its vey hard to imagine something which could be in some less excited state than minkowski.

You can argue that there should be no preferred background in a generally relativistic theory on the other hand there must be a ground state of the hilbert space. Flat Minkowski is the spacetime with the maximal conformal killing symmetries so it presents it's self as the most natural ground state.

The question then is to build general relativity with a small cosmological constant from string excitations. This would seem to involve not only gravitons but other modes of the superstring.

 

[tom.stoer]

The problem is that one should somehow categorize backgrounds in terms of something like "classes" or "superselection sectors". Different sectors may or may not be "connected" by dynamics. In string theory the specific background can affect the details of the degrees of freedom living on it.

I see the following problems:
- one has to identify the correct d.o.f. for each background (sector)
- there may be backgrounds (sectors) which cannot be equipped with a viable string theory
- dynamically connected backgrounds (sectors) cannot be studied coherently if they have different string d.o.f.

Now the question is how to construct viable string theories for certain classes of backgrounds relevant in GR, especially
- dynamical collaps, e.g. pre-Schwarzschild and pre-Kerr
- FRW, dS, ...

 

[Finbar]

My understanding is that these different "backgrounds" are actually different dynamical solutions to string theory corresponding to unstable minimum of the potential. These "backgrounds" are then themselves made up of some coherent state of stringy degrees of freedom that have been integrated out. It is then the low energy effective degrees of freedom that differ from one minimum to the next.

But for everyday string theory computations most of this is implicit and one just starts from a given background. This is because string theorists are often still working at low energies for which a background geometry should emerge.

 

[tom.stoer]

Think about a similar problem in QCD: you are not able to describe the phase transition between ordinary nucleon-matter and QGP using low-energy effective models chiral perturbtion theory based on pions, vector mesons etc. As long as you are within one "superselection sector" everything is fine, but as soon as you want to study the global picture it becomes difficult ...

... anyway, this is off-topic as the same superselction sectors exist in GR (FRW, dS, AdS, ...) and therefore to have a semiclassical / low-energy limit restricted to a certain sector is certainly sufficient (in the context of this thread). It's not a fundamental issue, it's simply time-consuming to formulate the different low-energy theories and show for each sector that GR does emerges.

 

[haushofer]

How surprised should we be that conformal invariance in ST gives as condition the Einstein vacuum field equations?

 

[suprised]

It's more surprising that this basic but non-trivial feature is apparently not known to many of those "critics".

 

[atyy]

How surprised should we be that conformal invariance in ST gives as condition the Einstein vacuum field equations?

I've read that Calabi's conjecture was not motivated by GR (although Yau's interest in it was). So maybe there's a "pure" reason for this.

 

[suprised]

I've read that Calabi's conjecture was not motivated by GR (although Yau's interest in it was). So maybe there's a "pure" reason for this.

No, "GR out of strings" has almost nothing to do with Calabi-Yau manifolds. It is a deep physical result.

 

[tom.stoer]

@suprised: Can you please explain which metrics are not compatible with or do not emerge from string theory? which conditions are required? (Ricci-flatness, static / stationary; Killing vectors, ...)? what about dS, FRW, ...?

 

[nrqed]

It's done in almost all the textbooks out there, in particular GSW. Online I think David Tong has some lecture notes: (see here: http://www.damtp.cam.ac.uk/user/tong/string/seven.pdf). The Einstein Hilbert action emerges on page 168. Alternatively I believe Susskind goes over it in his lectures on youtube (this will be at the level of Zweibach)

Actually calculating the full one loop beta functions is a bit of a chore, and I have never done it, but you will get the picture.

Tong says on page 158 that inserting a factor $$e^V$$ amounts to inserting a coherent state of gravitons (with V defined in his equation (7.2))
I have seen that statement many times before, of course. But I don't understand what it means. Is that in the usual sense of coherent state, i.e a state that, in the operator language, is an eigenstate of the annihilation operator? If so, how do we see that the exponential form in the path integral language corresponds to an eigenstate of the annihilation operator in the operator language?

 

[suprised]

@suprised: Can you please explain which metrics are not compatible with or do not emerge from string theory? which conditions are required? (Ricci-flatness, static / stationary; Killing vectors, ...)? what about dS, FRW, ...?

No I can't - metrics do not appear in isolation, there are other fields coupled to it, and only the whole package is consistent or not; so that's essentially a question about the swampland and there is no easy answer for that AFAIK.

 

[haushofer]

I've read that Calabi's conjecture was not motivated by GR (although Yau's interest in it was). So maybe there's a "pure" reason for this.

But the Einstein equations already pop up without compactification, right? It's only after this that one considers CY-compactification.

The first time I saw Einstein's equations popping up as a QM-consistency in string theory I was really impressed, especially in combination with the fact that the string spectrum contains gravitons. But I've never really understood how stringent this result is. Is it really "a deep physical result", or can it be understood more directly?

Perhaps a vague question, so never mind if it doesn't make sense ;)

 

[suprised]

But the Einstein equations already pop up without compactification, right? It's only after this that one considers CY-compactification.

Right.

The first time I saw Einstein's equations popping up as a QM-consistency in string theory I was really impressed, especially in combination with the fact that the string spectrum contains gravitons. But I've never really understood how stringent this result is. Is it really "a deep physical result", or can it be understood more directly?

One could vaguely say that by construction the effective action must be Lorentz invariant (the symmetry currents are conserved), so it is natural that GR pops out. But Lorentz invariance does not imply GR, eg one may contemplate about a theory with many massless higher spin fields or some other crazy theory. In fact crazy theories do pop out in certain singular limits (like tensionless strings), so that Einstein gravity emerges at all, in the limit where it is desired (small energies and curvature), seems nontrivial and I wouldn't know of a direct shortcut to prove this.

And there goes much more into that, eg the absence of anomalies which ensures conservation of symmetry currents also at the quantum level. That this actually appears to work in all detail is one of the main reasons for excitement.

 

[tom.stoer]

No I can't - metrics do not appear in isolation, there are other fields coupled to it, and only the whole package is consistent or not; so that's essentially a question about the swampland and there is no easy answer for that AFAIK.

Sounds OK, but why can one read something regarding Ricci-flatness? Is this just an oversimplification?

 

[suprised]

Ricci-flatness has to do with supersymmetry, ie this is a necessary condition for a compactification manifold for the existence of unbroken supercharges. That's written in all textbooks, eg GSW.

 

[tom.stoer]

So I got this completely wrong? I mean neither is there target space SUSY nor is the Universe Ricci-flat.

 

[Haelfix]

Tong says on page 158 that inserting a factor $$e^V$$ amounts to inserting a coherent state of gravitons (with V defined in his equation (7.2))
I have seen that statement many times before, of course. But I don't understand what it means. Is that in the usual sense of coherent state, i.e a state that, in the operator language, is an eigenstate of the annihilation operator? If so, how do we see that the exponential form in the path integral language corresponds to an eigenstate of the annihilation operator in the operator language?

Hi Nrqed, that is a truly excellent question and way above the level of Tong.

(for the identical statement said slightly differently, see Polchinksi):
http://books.google.com/books?id=k4...age&q=coherent states vertex operator&f=false

I think the answer to your first question is almost but not quite. The problem is there are gauge fixing ambiguities creeping into the calculation, and you have to ensure the symmetries of string theory (Virasoro constraints) are enforced. Consequently the naive definition of a coherent state must be slightly generalized to ensure this.

But once that is done, then yes there is a sense in which you can show that what you get is an eigenstate of the annihilation operator, although the paper I am looking at is technically challenging...

See
http://arxiv.org/abs/0911.5354v2 starting on page 27 for a discussion and the calculation for eg closed strings in lightcone gauge is on page 36, although the vertex operators are more general (DDF vertex operators). Maybe one of the stringy experts here knows a simpler calculation, but I have never seen it done nor could I find it in a quick literature search. I am actually a little surprised that I couldn't find the calculation done in texts regarding nonminimal sigma models, since this is very much isomorphic.

 

[Haelfix]

So I got this completely wrong? I mean neither is there target space SUSY nor is the Universe Ricci-flat.

The only thing you are guarenteed to get in low energy string theory (at least the usual constructions, where we are not talking about non critical s.t), are Einstein's equations, in particular (minimally) classical gravitational waves. You are most assuredly not guarenteed to get any sort of cosmological constant. In fact classically I believe there are no go theorems that preclude the existence of say a positive cosmological constant, thus that would need to be generated by quantum effects (for instance the KKLT solution). Very nontrivial!

As far as cosmological solutions. That is also a very difficult question, b/c you need to ensure a way that all the extra typical stringy stuff (gravitinos, dilatons, blah blah blah) doesn't forbid universes like our own, and you therefore need mechanisms so that you don't have too much of it floating around (otherwise it could say overclose the universe or mess with big bang nucleosynthesis constraints). There have been many proposals on how to do all this, and the literature and phenomenology is vast.